## (m-n)^6-8m^3: Factoring and Simplifying

This expression involves factoring a difference of cubes and then simplifying the resulting expression. Let's break down the steps:

### Step 1: Recognizing the Difference of Cubes Pattern

The expression (m-n)^6 - 8m^3 fits the pattern of a difference of cubes:

**a^3 - b^3 = (a-b)(a^2 + ab + b^2)**

In this case:

**a = (m-n)^2****b = 2m**

### Step 2: Applying the Difference of Cubes Formula

Applying the formula, we get:

(m-n)^6 - 8m^3 = **[(m-n)^2 - 2m][(m-n)^4 + 2m(m-n)^2 + 4m^2]**

### Step 3: Simplifying Further

**Simplify the first factor:**(m-n)^2 - 2m = m^2 - 2mn + n^2 - 2m**Expand the second factor:**(m-n)^4 + 2m(m-n)^2 + 4m^2 = m^4 - 4m^3n + 6m^2n^2 - 4mn^3 + n^4 + 2m^3 - 4m^2n + 2mn^2 + 4m^2

### Final Factored Expression

The fully factored and simplified expression is:

**(m^2 - 2mn + n^2 - 2m)[m^4 - 2m^3n + 6m^2n^2 - 4mn^3 + n^4 + 2m^3 - 4m^2n + 2mn^2 + 4m^2]**

This expression can't be factored further without using complex numbers. Therefore, this is the simplest form of the given expression.